We examine an electric double layer containing an antagonistic salt in an aqueous mixture, where the cations are small and hydrophilic but the anions are large and hydrophobic. In this situation, a strong coupling arises between the charge density σ and the solvent composition. As a result, the anions are trapped in an oil-rich adsorption layer on a hydrophobic wall. We then vary the surface charge density  on the wall. For σ > 0 the  anions remain accumulated, but for σ < 0 the cations are attracted to the wall with increasing |σ|. Furthermore, the electric potential drop Ψ(σ)  is nonmonotonic when the solvent interaction parameter χ(T)  exceeds a critical value χc determined by the composition and the ion density in the bulk. This leads to a first-order phase transition between two kinds of electric double layers with different σ and common Ψ . In equilibrium such two-layer regions can coexist. The steric effect due to finite ion sizes is crucial in these phenomena.